To solve for $\angle K$ in $\Delta JKL$, we can use the Law of Cosines. The law of cosines is given by:

$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$

In this case, we are given the sides $k = 840 \, \text{cm}$, $l = 950 \, \text{cm}$, and the angle $\angle J = 12^\circ$. To find $\angle K$, we'll first apply the Law of Sines:

$\frac{\sin J}{j} = \frac{\sin K}{k}$

However, we do not have side $j$, so let's first use the Law of Cosines to find the missing side $j$. After finding $j$, we'll apply the Law of Sines to calculate $\angle K$.

I'll proceed to calculate this.$angle K$ is approximately $54^\circ$ when rounded to the nearest degree.

Would you like further clarification on the steps or details about the process?

Here are some related questions to expand your understanding:

1. How is the Law of Cosines derived?
2. Can the Law of Sines be used when all angles are given?
3. How would the solution change if $\angle J$ were larger?
4. What other methods can be used to solve for unknown angles in triangles?
5. What are some practical applications of the Law of Sines?

Tip: Always double-check if the triangle is obtuse or acute, as it can affect angle rounding in sine and cosine rules.